Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 192156 11 pages doi 2011 192156 Research Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations Jian Zu College of Mathematics Jilin University Changchun 130012 China Correspondence should be addressed to Jian Zu zujian1984@ Received 22 May 2010 Accepted 6 March 2011 Academic Editor Kanishka Perera Copyright 2011 Jian Zu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We study periodic solutions for nonlinear second-order ordinary differential problem x f t x X 0. By constructing upper and lower boundaries and using Leray-Schauder degree theory we present a result about the existence and uniqueness of a periodic solution for second-order ordinary differential equations with some assumption. 1. Introduction The study on periodic solutions for ordinary differential equations is a very important branch in the differential equation theory. Many results about the existence of periodic solutions for second-order differential equations have been obtained by combining the classical method of lower and upper solutions and the method of alternative problems The Lyapunov-Schmidt method as discussed by many authors 1-10 . In 11 the author gives a simple method to discuss the existence and uniqueness of nonlinear two-point boundary value problems. In this paper we will extend this method to the periodic problem. We consider the second-order ordinary differential equation x f f x x J 0. Throughout this paper we will study the existence of periodic solutions of with the following assumptions Hl f fx and fxi are continuous in R X R X R and f f x x f f 2n x x 2 Boundary Value Problems H2 N2 a - Y- p N 1 2 sin nỵj 4a - Y2 4N IÌ 4a if N 0 1-3 Y 4 N 1 1 n p N 1
